Properties

Label 2-342-19.6-c1-0-0
Degree $2$
Conductor $342$
Sign $-0.911 - 0.411i$
Analytic cond. $2.73088$
Root an. cond. $1.65253$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (0.613 + 3.47i)5-s + (−1.85 + 3.21i)7-s + (0.500 + 0.866i)8-s + (−2.70 − 2.27i)10-s + (−2.64 − 4.58i)11-s + (0.213 − 0.0775i)13-s + (−0.645 − 3.66i)14-s + (−0.939 − 0.342i)16-s + (1.26 − 1.06i)17-s + (−4.17 + 1.24i)19-s + 3.53·20-s + (4.97 + 1.80i)22-s + (−1.50 + 8.54i)23-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (0.0868 − 0.492i)4-s + (0.274 + 1.55i)5-s + (−0.702 + 1.21i)7-s + (0.176 + 0.306i)8-s + (−0.855 − 0.717i)10-s + (−0.797 − 1.38i)11-s + (0.0590 − 0.0215i)13-s + (−0.172 − 0.978i)14-s + (−0.234 − 0.0855i)16-s + (0.307 − 0.257i)17-s + (−0.958 + 0.285i)19-s + 0.789·20-s + (1.05 + 0.385i)22-s + (−0.314 + 1.78i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.911 - 0.411i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.911 - 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.151263 + 0.703142i\)
\(L(\frac12)\) \(\approx\) \(0.151263 + 0.703142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 \)
19 \( 1 + (4.17 - 1.24i)T \)
good5 \( 1 + (-0.613 - 3.47i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (1.85 - 3.21i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.64 + 4.58i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.213 + 0.0775i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.26 + 1.06i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (1.50 - 8.54i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (0.0923 + 0.0775i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.56 - 2.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.12T + 37T^{2} \)
41 \( 1 + (-6.67 - 2.43i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.929 + 5.27i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-1.92 - 1.61i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.03 - 5.84i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (0.167 - 0.140i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.273 - 1.55i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-11.8 - 9.95i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.235 - 1.33i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (2.27 + 0.829i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-2.69 - 0.979i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.960 + 1.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (11.4 - 4.15i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-13.4 + 11.2i)T + (16.8 - 95.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.58215019917698502831836844913, −10.88585560460292664699669488760, −10.04112567287061910129562348075, −9.166036401201156171990535893676, −8.145329455899024109039407762610, −7.14334793917677641934845309528, −5.98637622238751697071979011601, −5.73300409824687349529713421674, −3.35044885545579258212897496685, −2.45475605806701639872563511289, 0.57462436625304100214247240492, 2.18546679350048505152936892426, 4.09808601343357153376842969770, 4.80178915581582679971186423751, 6.39989026969070823530672997864, 7.56578117959493767427840986670, 8.407314892965109838595100090189, 9.458900216059133308635500437055, 10.07749442700007218241534862321, 10.85621246250670903235210859041

Graph of the $Z$-function along the critical line